How can I make a conveyor belt FBX that spins without rigging?
If you want to make the belt "spin" without any rigging, I'd suggest animating the texture offset of the material applied to only the belt inside of Unity. 
This will give the illusion of the belt "spinnig"
http://docs.unity3d.com/ScriptReference/Material.SetTextureOffset.html

Just increase the x or y offset every frame in 

void Update ()


And make sure to multiply the increment by Time.deltaTime so the speed is consistent across all framerates